Complex Circuit Analysis

I'm working on a side project called a 'random circuit analyzer'. Essentially, with given information about randomly laid out conducting pieces (same material) across an area, I want to calculate the resistance from one end to the other (if there are any open paths).

The resistance of each piece varies with its length (its 1D). I've taken a courses in electromagnetism, so I'm aware of the Kirchoff's Laws and basic circuit analysis.

Before I start working on an algorithm, I was wondering if you guys do know any mathematical ways of analyzing complex circuits. There is only one power supply, and then there is a complex circuit (just many different paths). I was thinking of Fourier Series, but google didn't help much on the issue, so I thought I'd ask the experts.

You do not need to explain it computationally, just an explanation on how you would break down a complex circuit to find the resistance across it would be great.

Hi,
Fortunately, this is a simple one, if I'm understanding you correctly. No matter how many parallel resistors you have, the total resistance is:
Rtotal=1/[(1/R1)+(1/R2)+...+(1/Rn)], for a quantity of n resistors in parallel. That's what it sounds like you have. If you need the current, you can simply compute as follows: Itotal=(Vpwrsupply/Rtotal). I hope that helps. Your question was a bit cryptic and it sounded like you had a large number of parallel resistance paths. If it's more complicated than that, you can use KVL or KCL (Kirchoff's Voltage or Current laws) also known as Loop & Mesh analysis. Even though you said you were familiar with it, I'll say a bit about it for your benefit and for those who may not be familiar with it. No matter how complex or large a circuit is, it can be broken down into a series of sequential loops (for voltage analysis) or a series of nodes for current analysis. What you wind up with in either case is a series of simultaneous equations, which can be solved by a variety of means, such as reduction of variables (I can't remember the more common name), and most importantly, matrix analysis, which lends itself to solution by a computer algorithm. This is precisely how SPICE, PSPICE, HSPICE, etc, all work. The user inputs either a manual netlist after numbering each node, with node 0 always being the ground (return) node. For example, a simple parallel RC circuit might have a SPICE netlist as follows:
R1 1 0 1K
C1 1 0 1uF
where the R1, C1 is the component reference designator, the 1 0 means it connects from node 1 to node 0, and the 1K for the resistor means 1 KOHM. Actually, it doesn't matter what you put after the power-of-ten letter; the 2nd line could also have been written as: C1 1 0 1u or as
C1 1 0 1E-6. Obviously, most circuits would be more complex than this, but it serves as an example. SPICE then has an algorithm that turns the netlist into a set of simultaneous equations in matrix form, based on rules of KVL and/or KCL. More modern SPICE programs can allow the user to either input a manual netlist as above, or to draw a schematic using a schematic capture program that outputs the SPICE format netlist.
Good luck and I hope this helps.